Michael who has a fully HTTPS enabled site tried to apply but was rejected because his site was fully HTTPS. He posted his concern in the Google AdWords Help community and said:

In May, I was told that we did not qualify to become a trusted store at this time because our entire site is served via HTTPS, that you are aware of this limitation and that it is being addressed. Despite arguing that there is no reason to shuttle e-commerce customers between HTTP / HTTPS protocols anymore, that using HTTPS to protect all customer interactions is essential, and that Google now recognizes HTTPS as a signal to rank sites higher in search results, there was nothing that could be done and I would have to wait for a fix.

Google did indeed respond, where Mini from the AdWords team said they are working on it:

The Google Trusted Stores team is actively working with Merchants that have a full HTTPS website to make it possible for them to display the Google Trusted Stores badge. We ask that they please contact the Trusted Stores team for more information.

When will it come exactly? Well, that is unclear but I suspect very soon.